Sato-Tate Groups and Distributions of y^ℓ=x(x^ℓ-1)

2024/12/03 by Goodson, Heidi, Hoque, Rezwan
#11G10 #11G20 #14G10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.02522

Abstract

Let C_ℓ/\mathbb Q denote the curve with affine model y^ℓ=x(x^ℓ-1), where ℓ≥ 3 is prime. In this paper we study the limiting distributions of the normalized L-polynomials of the curves by computing their Sato-Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field \mathbb F_q satisfies q≡ 1 \pmodℓ².

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